Find the center of mass of a thin plate of constant density delta covering the given region. The region bounded by the parabola y = 3x - x^2 and the line y = -3x The center of mass is. (Type an ordered pair.)

Answer:

Let the length of the rectangle be l and the width be w. Then, according to the problem statement, we have:

Perimeter of rectangle = 2(l + w) = 48 feet

Dividing both sides by 2, we get:

l + w = 24 feet

Now we can check the options:

A) 6 feet x 8 feet: l + w = 6 + 8 = 14 feet, which is not equal to 24 feet. Therefore, this option is not correct.

B) 12 feet x 12 feet: l + w = 12 + 12 = 24 feet, which is equal to the given perimeter. Therefore, this option is correct.

C) 16 feet x 8 feet: l + w = 16 + 8 = 24 feet, which is equal to the given perimeter. Therefore, this option is correct.

D) 12 feet x 4 feet: l + w = 12 + 4 = 16 feet, which is not equal to 24 feet. Therefore, this option is not correct.

So, the correct options are B) 12 feet x 12 feet and C) 16 feet x 8 feet.

Answer:

None of the given options matches the value we got for x, but the closest option is A. 2 over 3. However, we need to note that x is not a whole number, it's a decimal.

Step-by-step explanation:

Let's solve the given equation

14x+312=2(12x+34)

Distribute the 2 on the right-hand side

14x+312=24x+68

Subtract 14x from both sides

312=10x+68

Subtract 68 from both sides

244=10x

Divide both sides by 10

x=24.4

Answer:

2 over 3

Step-by-step explanation:

none of the options match the value of x

Answer:

(c) is correct.

Step-by-step explanation:

(c) 9 + 2(4) = 9 + 8 = 17

The Z-score of the interval within standard deviations of the mean for a normal distribution contains 87% of the probability is 1.11 (rounded to two decimal places).

To find the z-score such that the interval within standard deviations of the mean for a normal distribution contains 87% of the probability, we need to use the standard normal distribution table (Z-table) or a calculator that has the inverse normal function.

The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of 1. It is denoted by the letter Z. Z-scores measure the number of standard deviations a data point is from the mean of the data set. A positive Z-score indicates a data point is above the mean, while a negative Z-score indicates a data point is below the mean.

To find the Z-score such that the interval within standard deviations of the mean for a normal distribution contains 87% of the probability, we first need to find the probability that is outside the interval. Since the interval is within standard deviations of the mean, we can use the empirical rule or the 68-95-99.7 rule to find the probability that is outside the interval.

The 68-95-99.7 rule states that 68% of the probability lies within 1 standard deviation of the mean 95% of the probability lies within 2 standard deviations of the mean 99.7% of the probability lies within 3 standard deviations of the mean. Since we are interested in the interval within standard deviations of the mean that contains 87% of the probability, we can assume that the interval is 1 standard deviation away from the mean.

Using the 68-95-99.7 rule, we can find the probability that is outside the interval:

100% - 68% = 32%

Since the probability that is outside the interval is 32%, we want to find the Z-score that corresponds to the probability of 16% on either side of the mean. We use the Z-table or a calculator that has the inverse normal function to find the Z-score that corresponds to a probability of 0.16.

From the Z-table, the Z-score that corresponds to a probability of 0.16 is 1.11 (rounded to two decimal places).

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In the following question, among the given options, Option (b) "Parabola B is wider than Parabola A" and option (d) "The line of symmetry of Parabola A is to the left of that of Parabola B" are the true statements.

The following statements are true about the parabolas: c. the parabolas have the same line of symmetry, and d. the line of symmetry of parabola A is to the right of that of parabola B.

Parabola A and Parabola B have the x-axis as the directrix, with the focus of Parabola A at (3,2) and the focus of Parabola B at (5,4). As the focus of Parabola A is to the left of the focus of Parabola B, the line of symmetry for Parabola A is to the right of the line of symmetry of Parabola B.

Parabola A and Parabola B may have different widths, depending on their equation, but this cannot be determined from the information given.

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The statement is incorrect due to the invalid assumption of f(x) = g(x) = 0, and the incorrect application of the comparison test.

The comparison test states that if 0 ≤ f(x) ≤ g(x) and the integral of g(x) dx diverges, then the integral of f(x) dx also diverges. However, the statement assumes that f(x) and g(x) are equal to zero, which means that 0 ≤ f(x) ≤ g(x) is not satisfied.

Additionally, the assumption that g(x) dx diverges does not necessarily imply that f(x) dx also diverges. For example, let g(x) = 1/x^2 and f(x) = 0 for all x. Then g(x) dx diverges, but f(x) dx converges to zero.

In conclusion, the statement is incorrect due to the invalid assumption of f(x) = g(x) = 0, and the incorrect application of the comparison test.

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Hence, if you merely move north or east on the grid map, there are 126 distinct ways to get to the Tower of London.

A way to arrange things or elements in a particular order is through permutation. In other terms, an orderly rearranging of a set of elements is referred to as a permutation. The symbol n! indicates how many different combinations there are for a set of n elements. Factorial, denoted by an exclamation mark, signifies dividing the number by all positive integers that are less than it by one. For instance, there are 4! = 4 x 3 x 2 x 1 = 24 permutations for a set of 4 items. In several branches of mathematics, including combinatorics, probability, and statistics, permutations are used.

given

You must go 4 blocks north and 5 blocks east to reach the castle. You must make a total of 9 moves to get to the castle because you can only go north or east (4 north and 5 east). Consider this to be a combination problem in which you must select 4 of the possible 9 moves to be in the north and the remaining 5 to be in the east.

Using the combination formula, we can write:

[tex]C(9,4) = 9! / (4! * (9-4)!) = 126[/tex]

Hence, if you merely move north or east on the grid map, there are 126 distinct ways to get to the Tower of London.

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Therefore, the side length of the frame is approximately 3.18 inches.

Length is a physical property that describes the distance between two points in space. It is a fundamental dimension in the study of geometry and is usually measured in units such as meters, centimeters, feet, or inches. In the context of mathematics, length can refer to the size or magnitude of a line segment, curve, or other geometric shape.

By the question.

To find the area of the stained-glass frame, we need to subtract the area of the mirror from the area of the larger square formed by the cut corners of the frame. Let's call the side length of each cut square "x".

The larger square formed by the cut corners has side length (3 + x + x) = (3 + 2x) since each cut square adds x inches to the original side length of the mirror.

The area of the larger square is then. [tex](3 + 2x)^{2}[/tex] = 9 + 12x + 4[tex]x^{2}[/tex]square inches.

The area of the mirror is[tex]3^{2}[/tex] = 9 square inches.

The area of the stained-glass frame is the difference between these two areas:

(9 + 12x + 4[tex]x^{2}[/tex]) - 9 = 12x + 4[tex]x^{2}[/tex]

We know that the area of the stained-glass frame is 91 square inches, so we can set this equal to the polynomial we just derived and solve for x:

12x + 4[tex]x^{2}[/tex] = 91

4[tex]x^{2}[/tex] + 12x - 91 = 0

We can use the quadratic formula to solve for x:

[tex]x= \frac{(-12±\sqrt{(12)^{2} -4*4(-91)}}{8}[/tex]

[tex]\frac{x= (-12±\sqrt{1480}}{8}[/tex]

We can discard the negative solution since we are looking for a positive length for the frame, so:

[tex]\frac{x= (-12±\sqrt{1480}}{8}[/tex]

x ≈ 3.18

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30 percent chance

There are 30 possible numbers that a player can get on their uniform. Out of these, there are 9 single-digit numbers (1, 2, 3, 4, 5, 6, 7, 8, and 9) and 21 double-digit numbers (10, 11, 12, ..., 29, 30).

If no two players can have the same number, then the probability that the first player will get a single-digit number is simply the number of single-digit numbers divided by the total number of possible numbers:

P(single-digit number) = 9/30 = 0.3

So the player is correct that there is a 30% chance that she will get a single-digit number on her uniform.

Answer: Exactly 26 of them are are home owners

Step-by-step explanation:

Answer:

it will take approximately 5.88 months for Don to pay off the remaining amount of $R = $85t = $85(5.88) = $499.80

Step-by-step explanation:

Don paid $500 upfront and will make monthly payments of $85 for the remaining amount. Let's assume the remaining amount he needs to pay is $R. The total cost of the furniture is the sum of the amount paid upfront and the remaining amount:

Total Cost = $500 + $R

Since he will be paying $85 per month, we can set up an equation to determine the time it will take to pay off the remaining amount:

$R = $85t

where t is the number of months it will take to pay off the remaining amount.

Substituting $R = $85t in the total cost equation, we get:

Total Cost = $500 + $85t

Since we want to find the time it will take to pay off the furniture, we need to solve for t. We can equate the total cost to the amount Don will pay at the end of the payment period, which is:

Total Cost = Amount Paid

$500 + $85t = $500 + $85t + $R

$85t = $R

$500 + $85t = $500 + $85t + $85t

$500 + $170t = $500 + $R

$170t = $R

Substituting $R = $85t, we get:

$170t = $85t

t = $500/$85

t = 5.88 (rounded to two decimal places)

Answer: 0.4

Step-by-step explanation:

I genuinly cant be asked to explain.

Answer:

To evaluate this expression, you need to follow the order of operations, which is:

Do any calculations inside parentheses first. (There are no parentheses in this expression.)

Exponents or radicals (There are no exponents or radicals in this expression.)

Multiplication or division, from left to right. (Perform 32 x 2.2, which equals 70.4.)

Addition or subtraction, from left to right. (Perform 12 divided by five, which equals 2.4, then add that to 70.4.)

Therefore, the answer is:

12 ÷ 5 + 32 x 2.2 = 2.4 + 70.4 = 72.8

The value οf x accοrding tο the circle and the tangent figure is 13 cm.

A tangent οn any curve is an extended straight line that tοuches οnly a single pοint οf the curve and nοwhere else

Tangent οn a circle is always perpendicular tο the radius οf the circle

Here, we have 2 tangents A (x) and B (13) subtended frοm twο pοints οf the same circle.

The Tangent οn a circle is perpendicular tο the radius thrοugh the pοint οf cοntact and thus the triangle fοrmed in the figure is right-angled.

Sο, frοm a pοint οutside the circle, if 2 tangents are drawn, bοth will have the same length tο the pοint οf cοntact οn the circle.

Here, the twο tangents have the same exteriοr pοint where the tangent initiates. Thus, frοm the abοve theοry, x = 13 cm.

Hence, the length οf the οther tangent tο the circle pοint οf cοntact i.e. x is 13 cm.

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Answer: 13

Step-by-step explanation:

x is congruent to 13

Answer: 3/5

hope this helped you. Please brainliest! :D

Step-by-step explanation: If I am wrong tell me :D

The value of x will be 9.

What is Transversal?

In geometry, a transversal is a line that intersects two or more other lines in a plane. When a transversal intersects two parallel lines, it creates eight angles, four on each side of the transversal. The angles that are opposite to each other and not next to each other are called alternate angles, while the angles that are on the same side of the transversal and not next to each other are called corresponding angles. The angles that are next to each other and on the same side of the transversal are called adjacent angles, and the angles that are opposite to each other and next to each other are called vertical angles.

We know that if two transversal cuts the parallel lines, then the ratio of length of corresponding sides is equal.

So, we have,

    8 / (x-3) = 4 / 3

Now, we can solve for x as follows:

  8 / (x-3) = 4 / 3

  8 × 3 = 4 (x-3)

  24 = 4x - 12

  24 + 12 = 4x

  36 = 4x

 ∴ x = 9.

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Answer:

Part B: Calculate the range and interquartile range (IQR) for each group and interpret what they tell us about the data.

For Group A:

Range = 5 - 1 = 4

Q1 = 2

Q3 = 4

IQR = Q3 - Q1 = 2

For Group B:

Range = 5 - 2 = 3

Q1 = 2

Q3 = 4

IQR = Q3 - Q1 = 2

The range for Group A is larger than the range for Group B, indicating that there is more variability in the growth of the plants in Group A. However, both groups have the same IQR, indicating that the middle 50% of the data in each group is similar. This suggests that while there may be some variability in the growth of the plants, the overall distribution of growth is similar between the two fertilizers.

Answer:

A ratio of 25 to 6 can be written as 25 to 6, 25:6, or 25/6. Furthermore, 25 and 6 can be the quantity or measurement of anything, such as students, fruit, weights, heights, speed and so on. A ratio of 25 to 6 simply means that for every 25 of something, there are 6 of something else, with a total of 31

Step-by-step explanation:

done hope u like it !!

Since the calculated t-value (3.14) is greater than the critical value (1.98), we reject the null hypothesis.

In statistical hypothesis testing, the null hypothesis is a statement about a population parameter that is assumed to be true until there is sufficient evidence to suggest otherwise. The null hypothesis is typically denoted by H0 and represents the status quo or default assumption.

The null hypothesis often takes the form of an equality or a statement of "no difference" or "no effect" between two or more groups, variables, or populations. For example, the null hypothesis could be that the mean score of a group of students on a test is equal to a certain value, or that there is no difference in the average height of males and females in a population.

We want to test the hypothesis that the mean amount of time spent in extracurricular activities per week is the same in the suburban and city school districts. Set up the null and alternative hypotheses is as given by:

Null hypothesis: u1 - u2 = 0

Alternative hypothesis: u1 - u2 ≠ 0

To test this hypothesis, we can use a two-sample t-test. We first calculate the test statistic:

t = ((x1 - x2) - (u1 - u2)) / √(s1²/n1 + s2²/n2)

where x1, s1, and n1 are the sample mean, standard deviation, and sample size for the suburban school district, and x2, s2, and n2 are the sample mean, standard deviation, and sample size for the city school district.

Plugging in the values, we get:

t = ((6 - 4) - 0) / √((3²/60) + (2²/40)) ≈ 3.14

This test's degrees of freedom are given by:

df = (s1²/n1 + s2²/n2)² / ( (s1²/n1)² / (n1 - 1) + (s2²/n2)² / (n2 - 1) )

Plugging in the values, we get:

df = ((3²/60) + (2²/40))² / ( (3²/60)² / 59 + (2²/40)² / 39 ) ≈ 93.24

Using a t-distribution table with 93 degrees of freedom and a level of significance of 0.05, we find the critical values to be approximately -1.98 and 1.98.

Since the calculated t-value (3.14) is greater than the critical value (1.98), we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean amount of time spent in extracurricular activities per week is different between the suburban and city school districts.

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Using the empirical rule, the percentage of the students have grade point averages that are between 1.42 and 3.7 is 99.7%

The empirical rule states that for a bell-shaped distribution, the percentage of data that lie within a specified number of standard deviations from the mean is as follows: 68% of the data lie within 1 standard deviation of the mean. 95% of the data lie within 2 standard deviations of the mean

99.7% of the data lie within 3 standard deviations of the mean. Mean = 2.56Standard Deviation = 0.38We want to know what percentage of students have a grade point average between 1.42 and 3.7. To do this, we need to convert 1.42 and 3.7 into standard deviations away from the mean.

Using the z-score formula:(1.42-2.56)/0.38 = -2.99 and(3.7-2.56)/0.38 = 3.00This tells us that a grade point average of 1.42 is about 2.99 standard deviations below the mean, and a grade point average of 3.7 is about 3 standard deviations above the mean.

Using the empirical rule, we know that 99.7% of the data lies within 3 standard deviations of the mean. So the percentage of students that have a grade point average between 1.42 and 3.7 is approximately 99.7%.Thus, the correct answer is 99.7%.

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Therefore, the total amount of money utilized on fuel in June 2022 is R11 095,60.

Percent is a way of expressing a quantity as a fraction of 100. It is denoted by the symbol %, which means "per hundred". Percentages are often used to represent proportions or ratios in various fields, including finance, science, and statistics. For example, an interest rate of 5% means that for every hundred dollars borrowed or invested, five dollars of interest will be charged or earned.

Here,

(a) To calculate the total distance covered by the water tanker in March 2022, we need to find the distance travelled per day and multiply it by the number of days in March.

Distance travelled per day = 2 × 18 = 36 km (since it's a return trip)

Number of weekdays in March = 31 - 4 (Saturdays) = 27

Total distance covered = distance per day × number of weekdays

= 36 km/day × 27 days

= 972 km

(b) To determine the quantity of fuel utilized by the water tanker in March 2022, we need to divide the total distance covered by the average fuel consumption rate.

Fuel consumption rate = 5 km/ℓ

Total distance covered = 972 km

Fuel utilized = total distance covered / fuel consumption rate

= 972 km / 5 km/ℓ

= 194.4 ℓ

(c) To determine the total amount of money utilized on fuel for the water tanker in March 2022, we need to multiply the fuel quantity by the fuel price.

Fuel price in March 2022 = R16,28/ℓ

Fuel utilized = 194.4 ℓ

Total cost of fuel = fuel price × fuel quantity

= R16,28/ℓ × 194.4 ℓ

= R3 163,39

Therefore, the total amount of money utilized on fuel for the water tanker in March 2022 is R3 163,39.

(d) To determine the total amount of money utilized on fuel in June 2022, we need to repeat the above calculation using the June fuel price.

Fuel price in June 2022 = R24,14/ℓ

Fuel capacity = 460 ℓ

Total cost of fuel = fuel price × fuel capacity

= R24,14/ℓ × 460 ℓ

= R11 095,60

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Complete question:

Records of the number of water tankers that were supplied to the construction site appear in the calender on ANNEXURE A. The water source is at a distance of about 18 km (return trip) from the construction site. The water tanker has a fuel capacity of 460 litres.. The rate of fuel consumption of the Mercedes water tanker averages 5 km/ℓ. The prices of fuel per litre in March and June 2022 appear below. JUNE 2022 FUEL PRICES \begin{tabular}{|l|l|} \hline DIESEL & COST \\ \hline 50ppm & R24,14 \\ \hline \end{tabular} Source: 4.1 (a) Calculate the total distance that the water tanker has covered in March (2) 2022. (b) Hence, determine the quantity of fuel that was utilized by the water tanker in March 2022. (c) Determine the total amount of money that was utilized on fuel for the water tanker in March 2022. (2) 4.2 Determine the total amount of money that was utilized on fuel in June 2022.

A  total of 750 people visited the natural history museum on Friday night.

The total number of people who visited the natural history museum on Friday night can be calculated by dividing the number of people who saw the dinosaur exhibit (165) by the percentage of visitors who saw the exhibit (22%).

To do this, we can use the following formula:

Total number of visitors = Number of visitors who saw the exhibit ÷ Percentage of visitors who saw the exhibit

Substituting the given values, we get:

Total number of visitors = 165 ÷ 0.22 = 750

Therefore, a total of 750 people visited the natural history museum on Friday night.

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1) The area of a circle with a diameter of 1 m is approximately 0.7854 square meters.

2) The area of a circle with a diameter of 100 cm is approximately 7853.98 square centimeters.

3) The area of a circle with a radius of 100 cm is approximately 314159.27 square centimeters.

4) The area of a circle with a radius of 500 mm is approximately 785398.16 square millimeters.

The formula for the area of a circle is A = πr², where A is the area and r is the radius of the circle.

1) If the diameter of the circle is 1 m, then the radius is 0.5 m. Therefore, the area of the circle is:

A = πr² = π(0.5)² = π(0.25) ≈ 0.7854 square meters.

2) If the diameter of the circle is 100 cm, then the radius is 50 cm. Therefore, the area of the circle is:

A = πr² = π(50)² = π(2500) ≈ 7853.98 square centimeters.

3) If the radius of the circle is 100 cm, then the area of the circle is:

A = πr² = π(100)² = π(10000) ≈ 314159.27 square centimeters.

4) If the radius of the circle is 500 mm, then the area of the circle is:

A = πr² = π(500)² = π(250000) ≈ 785398.16 square millimeters.

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Answer:

To find the equation of the line passing through the points (5,-12) and (0,-2), we first need to find the slope of the line:

slope = (change in y) / (change in x)

slope = (-2 - (-12)) / (0 - 5)

slope = 10 / (-5)

slope = -2

Now that we have the slope, we can use the point-slope form of the line equation to find the equation of the line:

y - y1 = m(x - x1)

where m is the slope, and (x1, y1) is one of the given points on the line.

Let's use the point (5,-12):

y - (-12) = -2(x - 5)

y + 12 = -2x + 10

y = -2x - 2

Therefore, the equation of the line passing through the points (5,-12) and (0,-2) is y = -2x - 2.

The intensity of the beam 17 feet below the surface is 0.440265 times the initial intensity i0 of the incident beam is I(17) ≈ 0.002678.

It can be calculated as:

Let I(t) be the intensity of the beam at a depth of t feet below the surface, and

let k be a constant of proportionality.

Then we have:

[tex]dI/dt = -kI[/tex]

This equation says that the rate of change of intensity with respect to depth is proportional to the intensity itself, and the negative sign indicates that intensity decreases as depth increases.

We can solve this differential equation using separation of variables:

[tex]dI/I = -k dt[/tex]

[tex]\int\ dI/I = \int\ -k dt[/tex]

[tex]ln(I) = -kt + C[/tex]

[tex]I = e^{(C - kt)}[/tex]

where C is the constant of integration.

Now we can use the given information to find the value of k and the constant of integration C.

We know that at a depth of 3 feet below the surface, the intensity is 25% of the initial intensity i0:

[tex]I(3) = 0.25 i0[/tex]

[tex]e^{(C - 3k)} = 0.25 i0[/tex]

We also know that the depth at which we want to find the intensity is 17 feet below the surface:

t = 17

Now we can use the equation we derived earlier to find the intensity at a depth of 17 feet:

[tex]I(17) = e^{(C - 17k)}[/tex]

To find the constant of integration C and the constant of proportionality k, we can use the fact that we have two equations with two unknowns. First, we can solve the equation for C:

[tex]e^{(C - 3k)} = 0.25 i0[/tex]

[tex]C - 3k = ln{(0.25 i0)}[/tex]

[tex]C = ln{(0.25 i0)} + 3k[/tex]

Now we can substitute this expression for C into the equation for I(17):

[tex]I(17) = e^{(C - 17k)}[/tex]

[tex]I(17) = e^{(ln(0.25 i0) + 3k - 17k)}[/tex]

[tex]I(17) = e^{(ln(0.25 i0) - 14k)}[/tex]

Finally, we can solve for k using the fact that we know the intensity decreases by a factor of 0.25 when the depth increases from 0 to 3 feet:

[tex]dI/dt = -kI[/tex]

[tex]ln(I) = -kt + C[/tex]

[tex]I(3) = 0.25 i0[/tex]

[tex]e^{(C - 3k)} = 0.25 i0[/tex]

Taking the natural logarithm of both sides, we have:

[tex]C - 3k = ln{(0.25 i0)}[/tex]

Substituting the expression for C we derived earlier, we have:

[tex]ln{(0.25 i0)} + 3k - 3k = ln{(0.25 i0)}[/tex]

[tex]ln{(0.25 i0)} = ln{(0.25 i0)}[/tex]

This equation is true for all values of k, so we can choose any value for k that satisfies the differential equation.

For simplicity, we can choose[tex]k = ln(4)/3[/tex], which makes the constant of proportionality equal to[tex]-ln(4)/3.[/tex]

Now we can substitute this value of k into our expression for I(17) and simplify:

[tex]I(17) = e^{(ln(0.25 i0) - 14k)}[/tex]

[tex]I(17) = e^{(ln(0.25 i0) - 14ln(4)/3)}[/tex]

[tex]I(17) = 0.25 i0 e^{(-14ln(4)/3)}[/tex]

[tex]I(17) \approx 0.002678[/tex]

The intensity of the beam 17 feet below the surface is approximately 0.002678.

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Note that the solution of the system of equations will be (-6, -2). (Option A)

A system of equations is a collection of two or more equations with a shared set of unknown variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.

Note that System of equations is represented by two straight lines on a graph.

And solution of the system of equations is the point of intersection of these lines, because that is the point where the values from both functions satisfy all the equations simultaneously.

From the graph attached, two straight lines represent the system of equations.

And the point of intersection of these lines is the solution.

Therefore, solution of the system of equations will be (-6, -2). (Option A)

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Full Question:

Although part of your question is missing, you might be referring to this full question:

What is the solution to the system of equations?
A (-6,2-)

B (-2, 6)

C (6,2)

D (-2,-6)​? See attached image.

The missing area or side length in the triangles are:

1: Area = 145 units²

2: Area =  17 units²

3: Area  = 29 units²

4: Area= 27 units²

5: length = √37 units

6: length = 2√26 units

7: length = 3√11 units

8: length  = 5√3 units

Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. That is:

c² = a² + b²

Where  a and b are the lengths of the legs, and c is the length of the hypotenuse

No. 1

Area (hypotenuse)  = 81 + 64 = 145 units²

No. 2

Area (hypotenuse)  = 16 + 1 = 17 units²

No. 3

Area (hypotenuse)  = 5² + 2² = 29 units²

No. 4

Area (leg)  = 36 - 9 = 27 units²

No. 5

length (hypotenuse)  = √(6² + 1²) = √37 units

No. 6

length (hypotenuse)  = √(10² + 2²) = 2√26 units

No. 7

length (leg)  = √(10² - 1²) = 3√11 units

No. 8

length (leg)  = √(10² - 5²) = 5√3 units

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Probability of getting heads when coin 1 is flipped= 0.4 Probability of getting heads when coin 2 is flipped= 0.7Number of flips= 10(a) To find: The probability that exactly 7 of the 10 flips land on headsWe can find the probability of getting the outcome with the combination of coin 1 and coin 2. P (coin 1 and heads) = P (coin 1) * P (heads|coin 1) = 0.5 * 0.4 = 0.2P (coin 2 and heads) = P (coin 2) * P (heads|coin 2) = 0.5 * 0.7 = 0.35.

Thus, the probability of getting heads= 0.2 + 0.35= 0.55Using the formula of binomial distribution= $^nC_x$ * p^x * (1 - p)^(n - x)Where n= 10, p= 0.55 and x= 7 We have, P (exactly 7 heads) = $^{10}C_7$ * (0.55)^7 * (1 - 0.55)^(10 - 7)= 120 * 0.0559 * 0.1664= 1.108%Thus, the probability that exactly 7 of the 10 flips land on heads is 1.108%.(b) To find: The conditional probability that exactly 7 of the 10 flips land on heads given that the first of these ten flips lands headsWe need to find the probability of getting the first head from coin 1 and coin 2.P (coin 1 and head) = P (coin 1) * P (head|coin 1) = 0.5 * 0.4 = 0.2P (coin 2 and head) = P (coin 2) * P (head|coin 2) = 0.5 * 0.7 = 0.35Thus, probability of getting the first head = 0.2 + 0.35= 0.55Using Bayes theorem, P (7 heads|1st head is heads) = P (1st head is heads|7 heads) * P (7 heads) / P (1st head is heads)P (1st head is heads|7 heads) = 1 (As we have already obtained the 7 heads)P (7 heads) = 0.01108 (As obtained in part a)P (1st head is heads) = P (coin 1 and head) + P (coin 2 and head)= 0.2 + 0.35= 0.55Thus, P (7 heads|1st head is heads) = 1 * 0.01108 / 0.55= 0.02216Thus, the conditional probability that exactly 7 of the 10 flips land on heads given that the first of these ten flips lands heads is 0.02216.

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The values of x such that the graph of f has a horizontal tangent are;x = 2π/3 + 2πn, 4π/3 + 2πn, where n is an integer.

1. The given derivative can be found by finding the first few derivatives and observing the pattern that occurs as shown below;Differentiating sin x with respect to x gives the derivative cos x. Continuing this process, the pattern that emerges is that sin x changes sign for every odd derivative, and stays the same for every even derivative. Therefore the 115th derivative of sin x can be expressed as follows;(d115/dx115)(sin x) = sin x, for n = 58 (where n is an even number)2. To find the values of x such that the graph of f has a horizontal tangent, we differentiate f with respect to x, and then solve for x such that the derivative equals zero. We have;f(x) = x + 2sin xDifferentiating f(x) with respect to x gives;f'(x) = 1 + 2cos xFor a horizontal tangent, f'(x) = 0, thus;1 + 2cos x = 02cos x = -1cos x = -1/2The solutions of the equation cos x = -1/2 are;x = 2π/3 + 2πn or x = 4π/3 + 2πnwhere n is an integer. Therefore the values of x such that the graph of f has a horizontal tangent are;x = 2π/3 + 2πn, 4π/3 + 2πn, where n is an integer.

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